1,216 research outputs found
Phaselocked patterns and amplitude death in a ring of delay coupled limit cycle oscillators
We study the existence and stability of phaselocked patterns and amplitude
death states in a closed chain of delay coupled identical limit cycle
oscillators that are near a supercritical Hopf bifurcation. The coupling is
limited to nearest neighbors and is linear. We analyze a model set of discrete
dynamical equations using the method of plane waves. The resultant dispersion
relation, which is valid for any arbitrary number of oscillators, displays
important differences from similar relations obtained from continuum models. We
discuss the general characteristics of the equilibrium states including their
dependencies on various system parameters. We next carry out a detailed linear
stability investigation of these states in order to delineate their actual
existence regions and to determine their parametric dependence on time delay.
Time delay is found to expand the range of possible phaselocked patterns and to
contribute favorably toward their stability. The amplitude death state is
studied in the parameter space of time delay and coupling strength. It is shown
that death island regions can exist for any number of oscillators N in the
presence of finite time delay. A particularly interesting result is that the
size of an island is independent of N when N is even but is a decreasing
function of N when N is odd.Comment: 23 pages, 12 figures (3 of the figures in PNG format, separately from
TeX); minor additions; typos correcte
The mathematics behind chimera states
Chimera states are self-organized spatiotemporal patterns of coexisting coherence and incoherence. We give an overview of the main mathematical methods used in studies of chimera states, focusing on chimera states in spatially extended coupled oscillator systems. We discuss the continuum limit approach to these states, Ott--Antonsen manifold reduction, finite size chimera states, control of chimera states and the influence of system design on the type of chimera state that is observed
Synchrony and bifurcations in coupled dynamical systems and effects of time delay
Dynamik auf Netzwerken ist ein mathematisches Feld, das in den letzten Jahrzehnten schnell gewachsen ist und Anwendungen in zahlreichen Disziplinen wie z.B. Physik, Biologie und Soziologie findet. Die Funktion vieler Netzwerke hĂ€ngt von der FĂ€higkeit ab, die Elemente des Netzwerkes zu synchronisieren. Mit anderen Worten, die Existenz und die transversale StabilitĂ€t der synchronen Mannigfaltigkeit sind zentrale Eigenschaften. Erst seit einigen Jahren wird versucht, den verwickelten Zusammenhang zwischen der Kopplungsstruktur und den StabilitĂ€tseigenschaften synchroner ZustĂ€nde zu verstehen. Genau das ist das zentrale Thema dieser Arbeit. ZunĂ€chst prĂ€sentiere ich erste Ergebnisse zur Klassifizierung der Kanten eines gerichteten Netzwerks bezĂŒglich ihrer Bedeutung fĂŒr die StabilitĂ€t des synchronen Zustands. Folgend untersuche ich ein komplexes Verzweigungsszenario in einem gerichteten Ring von Stuart-Landau Oszillatoren und zeige, dass das Szenario persistent ist, wenn dem Netzwerk eine schwach gewichtete Kante hinzugefĂŒgt wird. Daraufhin untersuche ich synchrone ZustĂ€nde in Ringen von Phasenoszillatoren die mit Zeitverzögerung gekoppelt sind. Ich bespreche die Koexistenz synchroner Lösungen und analysiere deren StabilitĂ€t und Verzweigungen. Weiter zeige ich, dass eine Zeitverschiebung genutzt werden kann, um Muster im Ring zu speichern und wiederzuerkennen. Diese Zeitverschiebung untersuche ich daraufhin fĂŒr beliebige Kopplungsstrukturen. Ich zeige, dass invariante Mannigfaltigkeiten des Flusses sowie ihre StabilitĂ€t unter der Zeitverschiebung erhalten bleiben. DarĂŒber hinaus bestimme ich die minimale Anzahl von Zeitverzögerungen, die gebraucht werden, um das System Ă€quivalent zu beschreiben. SchlieĂlich untersuche ich das auffĂ€llige PhĂ€nomen eines nichtstetigen Ăbergangs zu SynchronizitĂ€t in Klassen groĂer Zufallsnetzwerke indem ich einen kĂŒrzlich eingefĂŒhrten Zugang zur Beschreibung groĂer Zufallsnetzwerke auf den Fall zeitverzögerter Kopplungen verallgemeinere.Since a couple of decades, dynamics on networks is a rapidly growing branch of mathematics with applications in various disciplines such as physics, biology or sociology. The functioning of many networks heavily relies on the ability to synchronize the networkâs nodes. More precisely, the existence and the transverse stability of the synchronous manifold are essential properties. It was only in the last few years that people tried to understand the entangled relation between the coupling structure of a network, given by a (di-)graph, and the stability properties of synchronous states. This is the central theme of this dissertation. I first present results towards a classification of the links in a directed, diffusive network according to their impact on the stability of synchronization. Then I investigate a complex bifurcation scenario observed in a directed ring of Stuart-Landau oscillators. I show that under the addition of a single weak link, this scenario is persistent. Subsequently, I investigate synchronous patterns in a directed ring of phase oscillators coupled with time delay. I discuss the coexistence of multiple of synchronous solutions and investigate their stability and bifurcations. I apply these results by showing that a certain time-shift transformation can be used in order to employ the ring as a pattern recognition device. Next, I investigate the same time-shift transformation for arbitrary coupling structures in a very general setting. I show that invariant manifolds of the flow together with their stability properties are conserved under the time-shift transformation. Furthermore, I determine the minimal number of delays needed to equivalently describe the systemâs dynamics. Finally, I investigate a peculiar phenomenon of non-continuous transition to synchrony observed in certain classes of large random networks, generalizing a recently introduced approach for the description of large random networks to the case of delayed couplings
Conditions for wave trains in spiking neural networks
Spatiotemporal patterns such as traveling waves are frequently observed in
recordings of neural activity. The mechanisms underlying the generation of such
patterns are largely unknown. Previous studies have investigated the existence
and uniqueness of different types of waves or bumps of activity using
neural-field models, phenomenological coarse-grained descriptions of
neural-network dynamics. But it remains unclear how these insights can be
transferred to more biologically realistic networks of spiking neurons, where
individual neurons fire irregularly. Here, we employ mean-field theory to
reduce a microscopic model of leaky integrate-and-fire (LIF) neurons with
distance-dependent connectivity to an effective neural-field model. In contrast
to existing phenomenological descriptions, the dynamics in this neural-field
model depends on the mean and the variance in the synaptic input, both
determining the amplitude and the temporal structure of the resulting effective
coupling kernel. For the neural-field model we employ liner stability analysis
to derive conditions for the existence of spatial and temporal oscillations and
wave trains, that is, temporally and spatially periodic traveling waves. We
first prove that wave trains cannot occur in a single homogeneous population of
neurons, irrespective of the form of distance dependence of the connection
probability. Compatible with the architecture of cortical neural networks, wave
trains emerge in two-population networks of excitatory and inhibitory neurons
as a combination of delay-induced temporal oscillations and spatial
oscillations due to distance-dependent connectivity profiles. Finally, we
demonstrate quantitative agreement between predictions of the analytically
tractable neural-field model and numerical simulations of both networks of
nonlinear rate-based units and networks of LIF neurons.Comment: 36 pages, 8 figures, 4 table
Evoked Patterns of Oscillatory Activity in Mean-Field Neuronal Networks
Oscillatory behaviors in populations of neurons are oberved in diverse contexts. In tasks involving working memory, a form of short-term memory, oscillations in different frequency bands have been shown to increase across varying spatial scales using recording methods such as EEG (electroencephalogram) and MEG (magnetoencephalogram). Such oscillatory activity has also been observed in the context of neural binding, where different features of objects that are perceived or recalled are associated with one another. These sets of data suggest that oscillatory dynamics may also play a key role in the maintenance and manipulation of items in working memory.
Using similar recording techniques, including EEG and MEG, oscillatory neuronal activity has also been seen to occur when certain images that cause aversion and headaches in healthy human subjects or seizures in those with pattern-sensitive epilepsy are presented. The images most likely to cause such responses are those with dominant spatial frequencies near 3--5 cycles per degree, the same band of wavenumbers to which normal human vision exhibits the greatest contrast sensitivity.
We model these oscillatory behaviors using mean-field, Wilson-Cowan-type neuronal networks. In the case of working memory and binding, we find that including the activity of certain long-lasting excitatory synapses in addition to the usual inhibitory and shorter-term excitatory synaptic activity allows for bistability between a low steady state and a high oscillatory state. By coupling several such populations together, both in-phase and out-of-phase oscillations arise, corresponding to distinct and bound items in working memory, respectively. We analyze the network's dynamics and dependence on biophysically relevant parameters using a combination of techniques, including numerical bifurcation analysis and weak coupling theory. In the case of spatially resonant responses to static simtuli, we employ Wilson-Cowan networks extended in one and two spatial dimensions. By placing the networks near Turing-Hopf bifurcations, we find they exhibit spatial resonances that compare well with empirical results. Using simulations, numerical bifurcation analysis, and perturbation theory, we characterize the observed dynamics and gain mathematical insight into the mechanisms that lead to these dynamics
Chaotic exploration and learning of locomotion behaviours
We present a general and fully dynamic neural system, which exploits intrinsic chaotic dynamics, for the real-time goal-directed exploration and learning of the possible locomotion patterns of an articulated robot of an arbitrary morphology in an unknown environment. The controller is modeled as a network of neural oscillators that are initially coupled only through physical embodiment, and goal-directed exploration of coordinated motor patterns is achieved by chaotic search using adaptive bifurcation. The phase space of the indirectly coupled neural-body-environment system contains multiple transient or permanent self-organized dynamics, each of which is a candidate for a locomotion behavior. The adaptive bifurcation enables the system orbit to wander through various phase-coordinated states, using its intrinsic chaotic dynamics as a driving force, and stabilizes on to one of the states matching the given goal criteria. In order to improve the sustainability of useful transient patterns, sensory homeostasis has been introduced, which results in an increased diversity of motor outputs, thus achieving multiscale exploration. A rhythmic pattern discovered by this process is memorized and sustained by changing the wiring between initially disconnected oscillators using an adaptive synchronization method. Our results show that the novel neurorobotic system is able to create and learn multiple locomotion behaviors for a wide range of body configurations and physical environments and can readapt in realtime after sustaining damage
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Dynamics of Patterns
Patterns and nonlinear waves arise in many applications. Mathematical descriptions and analyses draw from a variety of fields such as partial differential equations of various types, differential and difference equations on networks and lattices, multi-particle systems, time-delayed systems, and numerical analysis. This workshop brought together researchers from these diverse areas to bridge existing gaps and to facilitate interaction
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